傳統投資組合理論主要的是以變異數來衡量風險，而其中又以Markowitz (1952)提出的平均數－變異數投資組合模型最為著名，在此模型中，由於共變異數矩陣的計算上較為困難且複雜，因此，Konno及Yamazaki (1991)另外提出了平均數－平均絕對離差模型，此模型不但節省了計算時間，並且在求解最適投資組合時，也不需要共變異數矩陣，所以降低了計算上的困難度。除此之外，亦有許多學者分別提出不同的風險測量方式，如Markowitz (1959)提出了半變異數(semivariance)的觀念，而Estrada (2008)即以此半變異數為損失風險的觀念發展出一種較簡易的平均數－半變異數模型；其次，Bawa及Lindenberg (1977)以左偏動差(lower partial moment)做為損失風險的觀念而發展出平均數－左偏動差模型；另外，Rockafellar及Uryasev (2000)則以條件風險值(conditional value-at-risk)為損失風險的觀念發展出平均數－條件風險值模型。綜觀上述不同風險測量之投資組合模型，本研究以半變異數、左偏動差、平均絕對離差、條件風險值來衡量投資組合的風險，與利用變異數來衡量風險作比較，分析其所求解出的最適投資組合之差異與進行相似度分析，文中發現在樣本內資料分析部分，MLPM與MSV之間的相似度指數位居第一，而MV與MMAD之間的相似度指數較高。

Traditionally, the measure of risk used in portfolio optimization models is the variance. Markowitz (1952) proposed the famous mean-variance (MV) model for the modern portfolio theory by defining variance as risk. However, a major obstacle in the application of the mean-variance model is the computational complexity of estimating the covariance matrix by the MV model. Thus, Konno and Yamazaki (1991) proposed the mean-mean absolute deviation (MMAD) as alternative to the mean variance (MV) model to avoid this shortcoming. On the other side, many researchers had proposed different points of view to measure risks in the portfolio theory. For example, Markowitz (1959) proposed another risk measurement, semivariance (SV). Recently, Estrada (2008) solved the optimization portfolio problems by evaluating the downside risk which is derived from the concept of the semivariance. Next, Bawa and Lindenberg (1977) developed a theory to evaluate the downside risk model named ”Mean Lower-Partial-Moment” (MLPM) model which is derived from the concept of the Lower Partial Moment. Finally, Rockafellar and Uryasev (2000) proposed ”Mean Conditional Value-at-Risk” (MCVaR) model which is derived from the concept of the Conditional Value-at-Risk (CVaR). The main subject of this paper is to find out the optimal portfolio by the comparison and analysis of the portfolio risk measured by variance, the portfolio risk measured by SV, the portfolio risk measured by LPM, the portfolio risk measured by CVaR, and the portfolio risk measured by MAD. Finally, we find that MLPM model has the higest similarity index with MSV model, and MV and MMAD model are more relative similar with each other.